Unit 2 - Functions
Required Prior Knowledge
Questions
Calculate Let \(f\left(x\right)=4x^{2}+3x-1\).
a) Evaluate \(f\left(1\right)\)
b) Determine the value of \(k\) if \(f\left(k\right)=0\)
Solutions
Get Ready
Questions
Find the remainder when \(x^{3}-11x+3\) is divided by \(x+5\).
Solutions
Notes
The Remainder Theorem states
When the polynomial \(P\left(x\right)\) is divided by \(x-k\) the remainder is given by \(P\left(k\right)\).
The can be generalised to
When the polynomial \(P\left(x\right)\) is divided by \(ax-b\) the remainder is given by \(P\left(\frac{b}{a}\right)\)
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Suppose we have a polynomial \(P\left(x\right)\). We can write \(P\left(x\right)\) as$$P\left(x\right)=Q\left(x\right)\times \left(ax+b\right)+R$$So $$P\left(\frac{b}{a}\right)=Q\left(\frac{b}{a}\right)\times \left(a\left(\frac{b}{a}\right)-b\right)+R$$But $$a\left(\frac{b}{a}\right)-b=0$$So$$P\left(\frac{b}{a}\right)=0+R=R$$
Examples and Your Turns
Example
Using the Remainder Theorem, find the remainder when \(x^{3}+2x+7\) is divided by \(\left(x+2\right)\).
Your Turn
Using the Remainder Theorem, find the remainder when \(x^{4}-3x^{3}+x-4\) is divided by \(\left(x+2\right)\).
Your Turn
When \(x^{3}-2x^{2}+ax+11\) is divided by \(\left(x-2\right)\) the remainder is \(1\). Find the value of \(a\).
Your Turn
When \(x^{3}-3x^{2}+kx+5\) is divided by \(\left(x+1\right)\) the remainder is \(-2\). Find the value of \(k\).
Your Turn
When \(x^{3}+x^{2}+ax+b\) is divided by \(x-2\) the remainder is \(20\), and when it is divided by \(x+5\) the remainder is \(6\). Find \(a\) and \(b\).
Your Turn
The polynomial \(Q\left(x\right)\) is such that$$Q\left(x\right)=\left(x-c\right)\left(x-2\right)^{2}$$for some constant \(c\).
Given that the remainder when \(Q\left(x\right)\) is divided by \(\left(x-1\right)\) is \(-4\), find the value of \(c\).
Notes
The Factor Theorem is a corollary of the remainder theorem.
It states
For any polynomial \(P\left(x\right)\) then \(\left(ax-b\right)\) is a factor of \(P\left(x\right)\) if and only if \(P\left(\frac{b}{a}\right)=0\).
In other words \(\frac{b}{a}\) is a root of the equation \(P\left(x\right)=0\).
That is, if \(2\) is a zero of \(P\left(x\right)\) then \(\left(x-2\right)\) is a factor of \(P\left(x\right)\), and vice versa.
The factor theorem is a powerful tool when working with polynomials, especially when we need to find the factorised form of a polynomial.
Examples and Your Turns
Example
Using the Factor Theorem, show that \(\left(2x-3\right)\) is a factor of \(2x^{3}-13x^2+19x-6\).
Your Turn
Show that \(\left(x-2\right)\) and \(\left(x+5\right)\) are factors of the polynomial \(f\left(x\right)=2x^{3}+13x^2+x-70\).
Example
Fully factorise \(x^{3}+3x^{2}-33x-35\).
Your Turn
Fully factorise \(x^{3}-6x^{2}-x+6\).
Your Turn
Find \(k\) given that \(\left(x-2\right)\) is a factor of \(x^{3}+kx^{2}-3x+6\).
Your Turn
\(x^{3}+x^{2}+ax+b\) has a factor of \(\left(x-1\right)\) and leaves a remainder of \(17\) when divided by \(\left(x-2\right)\). Find the constants \(a\) and \(b\).
Your Turn
The polynomial \(P\left(x\right)=2x^{3}+ax^{2}-5x+b\) has a remainder of \(10\) when divided by \(\left(x-1\right)\) and is exactly divisible by \(\left(x+2\right)\).
Find the values of \(a\) and \(b\).
Your Turn
The polynomial \(f\left(x\right)=ax^{3}+bx^{2}-17x+6\) has factors \(\left(x-3\right)\) and \(\left(x+1\right)\).
Find the values of \(a\) and \(b\), and hence determine the third factor of \(f\left(x\right)\).
Your Turn
A cubic polynomial \(P\left(x\right)\) has a leading coefficient of \(1\). It is known that \(P\left(2\right)=0\), \(P\left(-1\right)=0\) and when \(P\left(x\right)\) is divided by \(\left(x-3\right)\), the remainder is \(10\).
a) Find the expression for \(P\left(x\right)\) in the form \(ax^{3}+bx^{2}+cx+d\).
b) Hence, or otherwise, find the remainder when \(P\left(x\right)\) is divided by \(\left(x+3\right)\).
Your Turn
Given that \(\left(x-1\right)\) is a factor of \(P\left(x\right)=x^{3}-4x^{2}+mx+n\), and that \(P\left(x\right)\) has a remainder of \(30\) when divided by \(\left(x-4\right)\), find the values of \(m\) and \(n\).
Your Turn (GDC allowed)
Let \(g\left(x\right)=x^{4}+kx^{3}-2x^{2}+5x+10\) such that \(\left(x+2\right)\) is a factor of \(g\left(x\right)\).
a) Find the value of \(k\).
b) Hence, find all roots of \(g\left(x\right)\).
Your Turn
Prove that \(x-1\) is a factor of \(x^{n}-1\) for all \(n\in\mathbb{Z}^{+}\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Explain how the Factor Theorem is a corollary (a direct result) of the Remainder Theorem.
Does the Remainder Theorem still work if the divisor is not a linear functions (e.g. \(x^{2}+1\))? Explain why or why not.
Common Mistakes / Misconceptions
Using the wrong sign of the value, such as when the divisor is \(x-1\) working out \(f\left(-1\right)\).
Forgetting that if the divisor is a factor, then the remainder is \(0\).
Whilst not incorrect, it is usually more work to solve these problems using polynomial division than the remainder theorem.
Connecting This to Other Skills
This skill builds upon the ideas of Polynomials (2.19), and it requires a knowledge of Points of Interest (2.5).
You will often have to solve Simultaneous Equations (PK4) or even Systems of Equations (1.20) to find unknowns.
When we see the Conjugate Root Theorem and Fundamental Theorem of Algebra (2.21) we will combine them with the Remainder and Factor Theorems to solve more problems.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?